Slide 1 Problem of Coleman-Mazur on p-adic families of L-functions.pdf

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Slide 1 Problem of Coleman-Mazur on p-adic families of L-functions.pdf

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Slide 1 Problem of Coleman-Mazur on p-adic families of L-functions

Problem of Coleman-Mazur on p-adic families of L-functions A. A. Panchishkin? Institut Fourier, B.P.74, 38402 St.–Martin d’Hères, FRANCE e-mail : panchish@mozart.ujf-grenoble.fr, FAX: 33 (0) 4 76 51 44 78 Slide 1 Abstract For a prime number p ≥ 5, consider a primitive cusp eigenform f = fk of weight k ≥ 2, f = P∞ n=1 anq n, and consider a family of cusp eigenforms fk′ of weight k′ ≥ 2, k′ 7→ fk′ = P∞ n=1 an(k ′)qn, containing f for k′ = k, such that the Fourier coefficients an(k′) are given by certain p-adic analytic functions k′ 7→ an(k′) for (n, p) = 1, and let αp(k′) be a Satake p-parameter of fk′ . In The Eigencurve (1998), R.Coleman and B.Mazur stated the following problem: Given a prime p and a family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k′)) 0, to construct a two variable p-adic L-function interpolating on all k′ the Amice-Vélu p-adic L-functions Lp(fk′) studied in [Am-Ve] , [Vi76] and in [MTT]. A solution (2003, see [PaTV]) is described using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic Banach algebra A. ?A talk held in Kyoto on September 22, 2005, 10:30–12:00. Slide 2 Our p-adic L-functions are Mellin transforms of certain measures with values in A. We construct such measures from products of classical Eisenstein series, which produce distributions with values in certain Banach A-modules M = M(N ; A) of modular forms with coefficients in the algebra A. Another approach, based on modular symbols, was developed by Glenn Stevens. Applications of these results to the p-adic Birch and Swinnerton-Dyer conjecture were discussed by P.Colmez (Bourbaki talk, June 2003, [Colm03]). Contents 0 Statement of the problem of Coleman-Mazur 4 Slide 3 1 p-adic integration and the p-adic weight space 18 2 Coleman’s families 26 3 Main results 31 4 Construction of the admissible measure μ? 36 5 Criterion of admissibility 40 6 Modular Eisenstein distributions Φj 43 7 Algebraic A-linear form `α : MN (ψ;A)α → A 48